$\kappa$-Madness and Definability

TL;DR

This paper explores the set-theoretic landscape under the assumption of a supercompact cardinal, constructing a model where certain definability properties of mad families fail at an uncountable regular cardinal.

Contribution

It introduces a novel model demonstrating the non-existence of (_1())-kappa mad families assuming a supercompact cardinal.

Findings

Existence of a model with no _1()-kappa mad families

Utilizes large cardinal assumptions to control definability properties

Advances understanding of mad families at uncountable cardinals

Abstract

Assuming the existence of a supercompact cardinal, we construct a model where, for some uncountable regular cardinal $\kappa$, there are no $\Sigma^1_1(\kappa)-\kappa-$mad families.